Making Heads or Tails of Combined Landmark Configurations in GM data
Michael L. Collyer, Mark A. Davis, Dean C. Adams
6/8/2020
Imagine this scenario…
Digitizing landmarks comprising salamander heads and tails, on whole organisms.
Levis et al. (2016) Biological Journal of the Linnean Society, 118(3),569–581.
Imagine this scenario…
Produces a GPA result that looks like this:
Imagine this scenario…
But separate GPAs on heads and tails gives better results in terms of variation around individual landmarks!
Is there a way to combine Procrustes residuals from separate configurations for morphological analyses?
- When one might wish to do this
- How one might wish to do this
- Should separate configurations be weighted in combination?
- Should combined configurations be re-aligned with GPA?
- The
combine.subsets function in geomorph
- Parting thoughts
When one might wish to combine landmark configurations
- Moving structures
- Articulated
- Non-articulated
Adams (1999) introduced methods for (1) fixing the articulation angle (2D configurations) between separate configurations or (2) appending subsets of data. Vidal-García et al. (2018) extended the fixed-angle concept to 3D data (multiple points and planar rotations).
Adams (1999) Evolutionary Ecology Research, 1, 959–970; Vidal-García et al. (2018) Ecology and Evolution, 8(9), 4669-4675.
When one might wish to combine landmark configurations
- Moving structures
- Articulated
- Non-articulated
Non-articulated structures can be combined with the “separate subsets” method (Adams 1999). When combined, configurations should be scaled to relative sizes (GPA will render all configurations to unit size).
Davis et al. (2016) offered a simple way to do that with this equation:
\[CS^{'}_i=\frac{CS_{i}}{\sum_{i=1}^{g}CS_{i}},\]
where \(CS^{'}_i\) is the relative centroid size of configuration \(i\), which is a scalar multiplied by the coordinates when appending configurations. If one configuration is large and one is small, they will remain large and small in combination. (This is done per specimen.)
Note that combined configurations are not actually unit size, as Davis et al. (2016) suggested, but are consistently scaled across specimens.
Adams (1999) Evolutionary Ecology Research, 1, 959–970; Davis et al. (2016) PLoS ONE, 11(1), e0211753.
When one might wish to combine landmark configurations
- Moving structures
- Articulated
- Non-articulated
- Multiple planar views of a 3D object (Davis et al. 2016; Profico et al. 2020)
Profico et al. (2020) demonstrated that by combining multiple 2D configurations, it was possible to produce similar PC dispersion patterns to 3D configurations, which might be beneficial if 3D data collection is not easy or possible.
They also found issues with the Davis et al. (2016) approach and offered a new solution for relativizing centroid sizes (more details soon).
Davis et al. (2016) PLoS ONE, 11(1), e0211753; Profico et al. (2020) Hystrix,the Italian Journal of Mammalogy, 30, 157–165.
How one should combine landmark configurations
Collyer et al. (2020) proposed a general formula for obtaining relative centroid sizes
\[CS^{'}_i=\frac{w_iCS_{i}}{\sqrt{\sum_{i=1}^{g} \left(w_iCS_{i}\right)^2}},\]
where the denominator is the pooled centroid size from multiple configurations and \(w_i\) are a priori weights. Relative centroid sizes are then used to scale Procrustes residuals, \(\mathbf{Z}_i\), i.e.,
\[\mathbf{Z} = \begin{pmatrix}
CS^{'}_1\mathbf{Z}_1\\
CS^{'}_2\mathbf{Z}_2\\
\vdots\\
CS^{'}_g\mathbf{Z}_g\\
\end{pmatrix}.\]
\(\mathbf{Z}\) is a matrix of combined coordinates, centered at \(0,0\) (2D) or \(0,0,0\) (3D) with a (pooled) centroid size equal to \(1\).
If all \(w_i\) are equal (to \(1\)), we can call this relative centroid sizes via standard centroid size (\(SCS\)).
If \(w_i\) are not all equal, we can call this relative centroid sizes via weighted centroid size.
Collyer et al. (2020) Evolutionary Biology, in press.
How one should combine landmark configurations
Collyer et al. (2020) proposed a general formula for obtaining relative centroid sizes
\[CS^{'}_i=\frac{w_iCS_{i}}{\sqrt{\sum_{i=1}^{g} \left(w_iCS_{i}\right)^2}},\]
Profico et al. (2020) found the unweighted approach of Davis et al. (2016) – and by extension, when all \(w_i\) above are equal – had some flaws and offered a solution that
\[w_i = \left(p_{i}k \right)^{-1/2},\]
for the \(p\) landmarks in \(k\) dimensions. (\(k\) is not needed for comparing multiple centroid sizes in the same dimension.) These weights normalize centroid size (Dryden and Mardia 2016). Whereas centroid size finds the sum of squared distances of landmarks to their centroid, normalized centroid size finds the mean of squared distances.
Collyer et al. (2020) Evolutionary Biology, in press; Profico et al. (2020) Hystrix,the Italian Journal of Mammalogy, 30, 157–165; Dryden & Mardia (2016). Statistical shape analysis: With applications in R. Wiley.
How one should combine landmark configurations
Collyer et al. (2020) proposed a general formula for obtaining relative centroid sizes
\[CS^{'}_i=\frac{w_iCS_{i}}{\sqrt{\sum_{i=1}^{g} \left(w_iCS_{i}\right)^2}},\]
- Normalized centroid size might have comparative appeal, if one wishes to compare (or combine) sparse and dense configurations, especially if structures are not so disparate in size.
- Normalized centroid size might be more consistent with “anatomical size”.
- Note Profico et al. (2020) only used the numerator above as a solution, so relative centroid size does not range between 0 and 1, and combined configurations would be neither unit size nor consistent in size following combination.
- However, using the formula above, relative centroid size via standard centroid size (\(SCS\)) or via normalized centroid size (\(NCS\)) must range between 0 and 1 and will produce combined configurations, \(\mathbf{Z}\), with unit size.
Collyer et al. (2020) Evolutionary Biology, in press; Profico et al. (2020) Hystrix,the Italian Journal of Mammalogy, 30, 157–165.
Why normalize centroid size to find relative sizes of configurations?
As Profico et al. (2020) illustrated, circles with the same radius and surface area have different \(CS^{'}\) when using \(SCS\) but not when using \(NCS\) to relativize.
Notes
- \(\sqrt{0.707^2 + 0.707^2} = 1\) and \(\sqrt{0.302^2 + 0.953^2} = 1\)
- The configurations are not circles. They are isocagons, a decagon, and a hectogon.
Profico et al. (2020) Hystrix,the Italian Journal of Mammalogy, 30, 157–165.
Why normalize centroid size to find relative sizes of configurations?
This gives the impression that \(CS^{'}\) via \(NCS\) is independent of landmark density.
- Not so fast… more to come in a moment.
What about circles of different size?
What about configurations with interior and exterior landmarks (concentric circles)?
Why normalize centroid size to find relative sizes of configurations?
Why normalize centroid size to find relative sizes of configurations?
Notes
- When doubling the radius, \(CS^{'}\) via \(NCS\) is doubled. Does this make sense?
- When doubling the radius of a circle, the surface area increases \(4 \times\) (volume of a sphere would increase \(8 \times\)). Note that \(CS^{'}\) via \(SCS\) increases \(4 \times\).
- When adding an interior circle of landmarks \(CS^{'}\) via \(NCS\) was smaller despite equal “anatomical size”. Does this make sense?
- \(CS^{'}\) via \(SCS\) was larger, as it has to be by summing squared distances rather than averaging them.
Normalizing centroid size is not a universal solution
I.e., \(NCS\) will tend to make smaller objects larger in relative size, if landmark density is the same.
Normalizing centroid size is not a universal solution
We discuss non-uniformly distributed landmark distributions in Collyer et al. (2020), which only exacerbate issues.
Collyer et al. (2020) Evolutionary Biology, in press
Normalizing centroid size is not a universal solution
Empirical example
As a reminder
